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%I #13 Sep 26 2023 03:54:46
%S 1,1,1,1,2,2,1,3,6,5,1,4,13,22,14,1,5,23,67,90,42,1,6,36,156,381,394,
%T 132,1,7,52,305,1162,2307,1806,429,1,8,71,530,2833,9192,14589,8558,
%U 1430,1,9,93,847,5919,27916,75819,95235,41586,4862,1,10,118,1272,11070,70098,286632,644908,636925,206098,16796
%N Number array whose rows are the series reversions of x(1-x)/(1+x)^k, read by antidiagonals.
%C First row is the Catalan numbers A000108, second row is the large Schroeder numbers A006318, third row is A062992, fourth row is A007297. As a number triangle, this is T(n,k)=if(k<=n,sum{j=0..k, binomial((n-k)(k+1),k-j)*binomial(k+j,j)}/(k+1),0) with row sums A107112 and diagonal sums A107113.
%F T(n, k)=sum{j=0..k, binomial(n(k+1), k-j)*binomial(k+j, j)}/(k+1)
%e Array begins
%e 1,1,2,5,14,42,132,...
%e 1,2,6,22,90,394,1806,...
%e 1,3,13,67,381,2307,14589,...
%e 1,4,23,156,1162,9192,75819,...
%p A107111 := proc(n,k)
%p add(binomial(n*(k+1),k-j)*binomial(k+j,j),j=0..k);
%p %/(k+1) ;
%p end proc: # _R. J. Mathar_, Aug 02 2016
%t T[n_, k_] := Sum[Binomial[n (k + 1), k - j] Binomial[k + j, j], {j, 0, k}]/(k + 1);
%t Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 21 2020 *)
%Y Cf. A366012.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, May 12 2005
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